@b.gatessucks I'm curious why you didn't post that as an answer. I took the liberty of posting it for you, but I encourage you to post such things yourself in the future. If you wish to post it now, I'll be glad to delete mine.
–
Mr.Wizard♦Jan 14 '13 at 15:33

1

@Mr.Wizard It's not gonna be more helpful in an answer and it's quicker to add a comment. Thanks for taking the time.
–
b.gatessucksJan 14 '13 at 15:40

2

@b.gatessucks actually, it is: comments cannot be Accepted and voting is limited. It takes hardly a moment longer to make an answer than a comment, and you should get the credit for your own answers/ideas. Nevertheless, it's your prerogative.
–
Mr.Wizard♦Jan 14 '13 at 15:42

3

From the tutorial on tensors: "You can think of Inner as performing a "contraction" of the last index of one tensor with the first index of another. If you want to perform contractions across other pairs of indices, you can do so by first transposing the appropriate indices into the first or last position, then applying Inner, and then transposing the result back." For multiple contractions, as in the second example, transpose all involved indexes to the end, Flatten them into a single index, and then perform a contraction.
–
whuberJan 14 '13 at 15:58

3 Answers
3

If we transpose the indices of $v$ and $w$ so that $v'_{abe} = v_{aeb}$ and $w'_{ecd} = w_{ced}$, then we can compute $u = v' \cdot w'$:

u === Transpose[v, {1, 3, 2}] . Transpose[w, {2, 1, 3}]
(* True *)

We can use a similar trick to compute $q$ if we reorder $v_{dea} \to v''_{ade}$, except that this time the $d$ and $e$ indices in $v_{ade}''$ and $w_{deb}$ must treated as if they comprised a single index to be contracted. Flatten can do this for us:

The contraction is performed on the tensor product in which the first three indices belong to the factor v and the last three indices label w. Therefore, the indices corresponding to $e$ in your sum for u are in slots {2, 5}, and the two summations for q run over slots {1, 4} (for the variable $d$) and {2,5} (as in v).

No love for this answer? If it works it's nice. (I can't test it.)
–
Mr.Wizard♦Jan 15 '13 at 6:27

@Mr.Wizard Trust me, it works, it's the best. I consider this to be one of the most important news in ver.9, much more interesting than auto-completion tosh.
–
ArtesJan 15 '13 at 12:56

@Artes Thanks - and I still haven't gotten used to autocompletion; when it appears and I want to just navigate away by down-arrow, it annoyingly snags the cursor... I guess we have to take the good with the bad.
–
JensJan 15 '13 at 15:01

@Jens Since I've been getting used to autocompletion for 2 months, it seems quite natural and sometimes I lack it when using ver.8. Nevertheless I find it's a toy unlike the new tensor capabilites which are definitely still underestimated.
–
ArtesJan 17 '13 at 16:21

@Artes I think I'd like it better if autocomplete choices only appeared after pressing a certain special key (like Tab), not immediately while in the middle of a word. I find that intrusive.
–
JensJan 17 '13 at 17:06

Mathematica is a registered trademark of Wolfram Research, Inc. While the mark is used herein with the limited permission of Wolfram Research, Stack Exchange and this site disclaim all affiliation therewith.